# Shift operator

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{{short description|Linear mathematical operator which translates a function}}
{{About|shift operators in mathematics|operators in computer programming languages|Bit shift|the shift operator of group schemes|Verschiebung operator}}

In [mathematics](/source/mathematics), and in particular [functional analysis](/source/functional_analysis), the '''shift operator''', also known as the '''translation operator''', is an [operator](/source/Operator_(mathematics)) that takes a [function](/source/Function_(mathematics)) {{math|''x'' ↦ ''f''(''x'')}}
to its '''translation''' {{math|''x'' ↦ ''f''(''x'' + ''a'')}}.<ref>{{MathWorld|id=ShiftOperator|title=Shift Operator}}</ref> In [time series analysis](/source/time_series_analysis), the shift operator is called the ''[lag operator](/source/lag_operator)''.

Shift operators are examples of [linear operator](/source/linear_operator)s, important for their simplicity and natural occurrence. The shift operator action on [functions of a real variable](/source/Function_of_a_real_variable) plays an important role in [harmonic analysis](/source/harmonic_analysis), for example, it appears in the definitions of [almost periodic functions](/source/almost_periodic_function), [positive-definite function](/source/positive-definite_function)s, [derivative](/source/derivative)s, and [convolution](/source/convolution).<ref name=mar>{{cite book|mr=2182783|last=Marchenko|first=V. A.|author-link=Vladimir Marchenko|chapter=The generalized shift, transformation operators, and inverse problems|title=Mathematical events of the twentieth century|pages=145&ndash;162|publisher=Springer|location=Berlin|year=2006|doi=10.1007/3-540-29462-7_8|isbn=978-3-540-23235-3 }}</ref> Shifts of sequences (functions of an [integer](/source/integer) variable) appear in diverse areas such as [Hardy space](/source/Hardy_space)s, the theory of [abelian varieties](/source/abelian_variety), and the theory of [symbolic dynamics](/source/symbolic_dynamics), for which the [baker's map](/source/baker's_map) is an explicit representation. The notion of [triangulated category](/source/triangulated_category) is a [categorified](/source/categorification) analogue of the shift operator.

==Definition==

===Functions of a real variable===

The shift operator {{mvar|T<sup> t</sup>}} (where {{tmath|t \in \R}}) takes a function {{mvar|f}} on {{tmath|\R}} to its translation {{mvar|f<sub>t</sub>}}, 
: <math>T^t f(x) =  f_t(x) = f(x+t)~.</math>

A practical [operational calculus](/source/operational_calculus) representation of the linear operator {{mvar|T<sup> t</sup>}} in terms of the plain derivative {{tmath|\tfrac{d}{dx} }} was introduced by [Lagrange](/source/Lagrange),

{{Equation box 1
|indent =:
|equation = <math>T^t= e^{t \frac d {dx}}~, </math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}

which may be interpreted operationally through its formal [Taylor expansion](/source/Taylor_expansion) in {{mvar|t}}; and whose action on the monomial {{mvar|x<sup>n</sup>}} is evident by the [binomial theorem](/source/binomial_theorem), and hence  on ''all series in'' {{mvar|x}}, and so all functions {{math|''f''(''x'')}} as above.<ref>Jordan, Charles, (1939/1965). ''Calculus of Finite Differences'', (AMS Chelsea Publishing), {{isbn|978-0828400336}} .</ref> This, then, is a formal encoding of the Taylor expansion in Heaviside's calculus.

The operator thus provides the prototype<ref>[M Hamermesh](/source/Morton_Hamermesh) (1989), ''Group Theory and Its Application to Physical Problems'' 
(Dover Books on Physics), Hamermesh ISBM 978-0486661810, Ch 8-6, pp 294-5, 
[https://physics.stackexchange.com/questions/331635/undefined-phase-flow/331841#331841 online].</ref> 
for Lie's celebrated [advective flow for Abelian groups](/source/Iterated_function),
:<math> \exp\left(t \beta(x) \frac{d}{dx}\right) f(x) = \exp\left(t \frac{d}{dh}\right) F(h) = F(h+t) = f\left(h^{-1}(h(x)+t)\right),</math>
where the canonical coordinates {{mvar|h}} ([Abel functions](/source/Abel_equation)) are defined such that 
:<math>h'(x)\equiv \frac 1 {\beta(x)} ~, \qquad f(x)\equiv F(h(x)). </math>

For example, it easily follows that  <math>\beta (x)=x</math> yields scaling,
:<math> \exp\left(t x \frac{d}{dx}\right) f(x) =   f(e^t x) , </math>
hence <math> \exp\left(i\pi x \tfrac{d}{dx}\right) f(x) = f(-x)</math> (parity); likewise, 
<math>\beta (x)=x^2</math> yields<ref>p 75 of  Georg Scheffers (1891): ''Sophus Lie, Vorlesungen Ueber Differentialgleichungen Mit Bekannten Infinitesimalen Transformationen'', Teubner, Leipzig, 1891. {{isbn|978-3743343078}}  [https://books.google.com/books?id=7-86AQAAIAAJ&q=+75&pg=PR6 online]
</ref> 
:<math> \exp\left(t x^2 \frac{d}{dx}\right) f(x) = f \left(\frac{x}{1-tx}\right),</math>

<math>\beta (x)= \tfrac{1}{x}</math> yields 
:<math> \exp\left(\frac{t} {x} \frac{d}{dx}\right) f(x) = f \left(\sqrt{x^2+2t} \right) ,</math>

<math>\beta (x)=e^x</math> yields 
:<math> \exp\left (t e^x \frac d {dx} \right ) f(x) = f\left (\ln \left (\frac{1}{e^{-x} - t} \right ) \right )  ,</math>
etc.

The [initial condition](/source/initial_condition) of the flow and the group property completely determine the entire Lie flow, providing a solution to the translation functional equation<ref  name="acz">Aczel, J (2006), ''Lectures on Functional Equations and Their Applications'' (Dover Books on Mathematics, 2006), Ch. 6, {{isbn|978-0486445236}} .</ref>
:<math>f_t(f_\tau (x))=f_{t+\tau} (x)  .</math>

===Sequences===
{{main|Unilateral shift operator}}

The '''left shift''' operator acts on one-sided [infinite sequence](/source/infinite_sequence) of numbers by

:<math> S^*: (a_1, a_2, a_3, \ldots) \mapsto (a_2, a_3, a_4, \ldots)</math>

and on two-sided infinite sequences by

:<math> T: (a_k)_{k\,=\,-\infty}^\infty \mapsto (a_{k+1})_{k\,=\,-\infty}^\infty.</math>

The '''right shift''' operator acts on one-sided [infinite sequence](/source/infinite_sequence) of numbers by

:<math> S: (a_1, a_2, a_3, \ldots) \mapsto (0, a_1, a_2, \ldots)</math>

and on two-sided infinite sequences by

:<math> T^{-1}:(a_k)_{k\,=\,-\infty}^\infty \mapsto (a_{k-1})_{k\,=\,-\infty}^\infty.</math>

The right and left shift operators acting on two-sided infinite sequences are called '''''bilateral''''' shifts. A finite-dimensional analog is given by the [shift matrices](/source/shift_matrix).

===Abelian groups===

In general, as illustrated above, if {{mvar|F}} is a function on an [abelian group](/source/abelian_group) {{mvar|G}}, and {{mvar|h}} is an element of {{mvar|G}}, the shift operator {{mvar|T<sup> g</sup>}} maps {{math|''F''}} to<ref name="acz" /><ref>"A one-parameter continuous group is equivalent to a group of translations". M Hamermesh, ''ibid''.</ref>
:<math> F_g(h) = F(h+g).</math>

==Properties of the shift operator==

The shift operator acting on real- or complex-valued functions or sequences is a linear operator  which preserves most of the standard [norms](/source/norm_(mathematics)) which appear in functional analysis. Therefore, it is usually a [continuous operator](/source/continuous_operator) with norm one.

===Action on Hilbert spaces===

The shift operator acting on two-sided sequences is a [unitary operator](/source/unitary_operator) on  {{tmath|\ell_2(\Z).}} The shift operator acting on functions of a real variable is a unitary operator on {{tmath|L_2(\R).}}

In both cases, the (left) shift operator satisfies the following commutation relation with the [Fourier transform](/source/Fourier_transform):
<math display="block"> \mathcal{F} T^t = M^t \mathcal{F}, </math>
where {{mvar|M<sup> t</sup>}} is the [multiplication operator](/source/multiplication_operator) by {{math|exp(''itx'')}}. Therefore, the spectrum of {{mvar|T{{sup| t}}}} is the [unit circle](/source/unit_circle).

The one-sided shift {{mvar|S}} acting on {{tmath|\ell_2(\N)}} is a proper [isometry](/source/isometry) with [range](/source/range_of_a_function) equal to all [vectors](/source/Vector_(geometric)) which vanish in the first [coordinate](/source/coordinate). The operator {{mvar|S}} is a [compression](/source/compression_(functional_analysis)) of {{math|''T''{{i sup|−1}}}}, in the sense that
<math display="block">T^{-1}y = Sx \text{ for each } x \in \ell^2(\N),</math>
where {{mvar|y}} is the vector in {{tmath|\ell_2(\Z)}} with {{math|1=''y<sub>i</sub>'' = ''x<sub>i</sub>''}} for {{math|''i'' &ge; 0}} and {{math|1=''y<sub>i</sub>'' = 0}} for {{math|''i'' < 0}}. This observation is at the heart of the construction of many [unitary dilation](/source/unitary_dilation)s of isometries.

The [spectrum](/source/Spectrum_(functional_analysis)) of {{mvar|S}} is the [unit disk](/source/unit_disk). The shift {{mvar|S}} is one example of a [Fredholm operator](/source/Fredholm_operator); it has Fredholm index&nbsp;−1.

==Generalization==

[Jean Delsarte](/source/Jean_Delsarte) introduced the notion of '''generalized shift operator''' (also called '''generalized displacement operator'''); it was further developed by [Boris Levitan](/source/Boris_Levitan).<ref name = mar/><ref>{{SpringerEOM|id=g/g043800|first=B.M.|last=Levitan|author-link=Boris Levitan|first2=G.L.|last2=Litvinov|title=Generalized displacement operators}}</ref><ref>{{SpringerEOM|id=A/a011970|first=E.A.|last= Bredikhina|title=Almost-periodic function}}</ref>

A family of operators {{tmath|\{L^x\}_{x \in X} }} acting on a space {{math|Φ}} of functions from a set {{mvar|X}} to {{tmath|\C}} is called a family of generalized shift operators if the following properties hold:
# [Associativity](/source/Associative_property): let <math>(R^y f)(x) = (L^x f)(y).</math> Then <math>L^x R^y = R^y L^x.</math>
# There exists {{mvar|e}} in {{mvar|X}} such that {{mvar|L<sup>e</sup>}} is the [identity operator](/source/Identity_function).
In this case, the set {{mvar|X}} is called a [hypergroup](/source/hypergroup).

==See also==
*[Arithmetic shift](/source/Arithmetic_shift)
*[Logical shift](/source/Logical_shift)
*[Clock and shift matrices](/source/Clock_and_shift_matrices)
*[Finite difference](/source/Finite_difference)
*[Translation operator (quantum mechanics)](/source/Translation_operator_(quantum_mechanics))

==Notes==
{{Reflist}}

==Bibliography==
* {{cite book | last=Partington | first=Jonathan R. | title=Linear Operators and Linear Systems | publisher=Cambridge University Press | date=March 15, 2004 | isbn=978-0-521-83734-7 | doi=10.1017/cbo9780511616693}}
* Marvin Rosenblum and James Rovnyak, ''Hardy Classes and Operator Theory'', (1985) Oxford University Press.

{{DEFAULTSORT:Shift Operator}}
Category:Unitary operators

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Adapted from the Wikipedia article [Shift operator](https://en.wikipedia.org/wiki/Shift_operator) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Shift_operator?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.